Abstract: There is an explosion of interest and demand to analyze data sampled on graphs and networks, thanks to the advent of new sensor technologies and social network infrastructure, as evidenced by many recent special conferences and publications of journal special issues. Yet, mathematical and computational tools for analyzing such datasets, particularly for those on directed graphs, have not been well developed. Conventional harmonic analysis tools such as Fourier and wavelet transforms as well as multiscale basis dictionaries, e.g., wavelet packets and local trigonometric transforms, have a proven track record of success in a variety of applications for functions supported on simple Euclidean domains and data sampled on regular lattices.

I will describe our effort of building multiscale basis dictionaries and best bases selected from such dictionaries for graphs and networks. In particular, I will describe two of such dictionaries: Hierarchical Graph Laplacian Eigen Transform (HGLET) and the Generalize Haar-Walsh Transform (GHWT). These are the generalization of the local cosine transform and the Haar-Walsh wavelet packets on the regular lattices. I will then demonstrate their usefulness by examining their performance on a variety of data analysis tasks on graphs and networks such as denoising, signal segmentation, and matrix data analysis.